# Full text of "Treatise On Analysis Vol-Ii"

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```21    PRODUCT OF MEASURES       231

(b)    Show  that  for / to   be  ^t-integrable  it   is   necessary   and   sufficient  that
D/ should be v-integrable, and that we  have then  v(D/) =   /'dp.   Moreover,
if g is the decreasing real-valued function on R+ defined by g(t) = p*(f~l [t, +00]),
then (fdjjL = [** g(t)dt.

(c)    Suppose that X is an interval [0, a] (where a > 0) and take ft to be Lebesgue
measure on X. If/is a decreasing /x-integrable function, show that

/•                 /^<A>

f(t)dt<,         f(t)dt

JA               Jo

for every /x-measurable subset A of X. (Use (b).)

L (a) With the notation of Problem 1, let Tf be the graph of/, i.e., the set of all points
(#,/(*)) in X X R. If/is /z-measurable, show that Pf is v-negligible (use (13.21.13)
and (13.21.10)).

(b) Let 0 < a < 1 and let Q(a) be the subset of R2 which is the complement of the
union of ]-l,l[x]— a,a[and] — a, a[x ] —1,1 [in the squareQ = [-1,1] X [-1,1].
The set Q(a) is the union of its four connected components

Ch(a) = [-1, -a] X [-1, -a],       Q2(a) = [-1, -a] X [a, 1],
Q3(a)= [a, 1] X [a, 1],                    Q4(a) = [a, 1] X [-1, -a].

For i = 1, 2, 3,4 let hit a denote the similitude mapping Q to Qi(a) for which

/h,a(-l, -!) = (-!, -1),      Alf.(l, -!) = (-!, -a),

h2, «(-!, -0 = (-1, a),         h2t .(1, -1) - (-a, a),

h3.«(-!, -1) = («, a),           h3,8(1, -1) = (1, a),

/U. .(-1, ™ D = (1, -a),         V a(l, -1) = (1, -1).

Also, in the interval [0, 7] in R, put Ifc = [A, k -f 1 ] for 0 g k <; 6 and let w* be
the increasing similitude which maps [0, 7] to I*.

Let/a be the continuous mapping of [0, 7] into Q which is affine-lineaf on each
of the intervals I* and is such that the images of 0, 1, 2,..., 7 are respectively the
points (-1, -1), (-1, -a), (-1, a), (-a, a), (a, a), (1, a), (1, -a), (1, -1).

We shall now define a sequence (gn) of continuous mappings of [0, 7] into Q as
follows. Let (an)n>0 be a decreasing sequence of numbers belonging to the interval
]0,1 [. Define g0 ==/«0. For n S> 1, suppose that gn~i has been defined, and consider
all sequences s = (j\,..., /„) in which each 4 is one of the integers 0, 1,..., 6. Put
t;s = Ull o • • - Q uin. Then it is sufficient to define gn(vs(t)) for 0 ^ / <; 7 and for each
of the 7" sequences s. If at least one of the /i is odd, we put ^(^(O) = #n-i(us(0)-
If on the other hand /t = 2jt for 1 g / <i n (with 0 <>ji g 3), we put#w(t>s(/))« w,(/ai>(0),
where H>S = hJl +1, ai ° hj2 + it a2 <= • • • o hjn+i( «„. Show that the sequence (gn) converges
uniformly to an injective continuous mapping g : [0, 7] -> R2, and that the simple arc
g([Q, 7)) (Section 4, Appendix to Chapter IX) is nonnegligible with respect to Lebesgue
measure on R2, if the sequence (an) is suitably chosen.

L (a) Give an example of two compact spaces X and Y, positive measures A and fju
on X and Y, respectively, and a A ® ^-measurable function / such that the two

integrals djj,(y) I /(;c, y) d\(x) and I dX(x) f(x, y) d^(y) are both defined and are
unequal (cf. Section 5.2. Problem 5).b)    Let  x0eX.   For  the series £(#|/,)/n(#o)  to  converge for  every  function
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